Lemma 8.2 ([1]})
Suppose (A1-A2, B2) regularity conditions hold. Denote \(f_n(X)\)  and \(f_{n,-j}(X_{-j})\)  as the estimate for \(f_{0}\)  and \(f_{0,-j}\) ,
Then for a predictive skill measure \(V(f,P)\)  satisfying conditions (A1)-(A2), (B2) in Appendix, as long as the estimators satisfy the following condition:
\(\Vert  f_n - f_0\Vert _{\mathcal {F}} = O_p(n^{-\frac{1}{4}}), ~~\Vert  f_{n,-j} - f_{0,-j}\Vert _{\mathcal {F}} = O_p(n^{-\frac{1}{4}}),\) 
